Robust project management with the tilted beta distribution

Statistics & Operations Research Transactions

SORT 39 (2) July-December 2015, 253-272

Statistics &Operations Research

Transactions© Institut d’Estadıstica de Catalunya

[email protected]: 1696-2281eISSN: 2013-8830www.idescat.cat/sort/

Robust project management with the

tilted beta distribution

Eugene D. Hahn1 and Marıa del Mar Lopez Martın2

Abstract

Recent years have seen an increase in the development of robust approaches for stochastic

project management methodologies such as PERT (Program Evaluation and Review Technique).

These robust approaches allow for elevated likelihoods of outlying events, thereby widening

interval estimates of project completion times. However, little attention has been paid to the fact

that outlying events and/or expert judgments may be asymmetric. We propose the tilted beta

distribution which permits both elevated likelihoods of outlying events as well as an asymmetric

representation of these events. We examine the use of the tilted beta distribution in PERT with

respect to other project management distributions.

MSC: 62E15, 90B99, 62P30

Keywords: Activity times, finite mixture, PERT, tilted beta distribution, robust project management,

sensitivity analysis.

1. Introduction

In project management it is important to be able to assess the total time for a project’s

completion. Since projects can be very complex, methodologies such as the Program

Evaluation and Review Technique (PERT) (Malcolm et al., 1959) have been developed

to assist in these assessments. PERT has been used for many decades but in recent years

academics, managers, and policy makers have increasingly realized that conventional

modeling approaches and tools may not be well-equipped to deal with extreme events.

For example, few lenders would have predicted that the rise of lending to the sub-prime

1 Department of Information and Decision Sciences, Salisbury University, Salisbury, MD 21801 USA. [email protected]

2 Department of Didactics of Mathematics. Campus Universitario Cartuja, s/n. 18071. University of Granada,Spain. [email protected]

Received: March 2015

Accepted: June 2015

254 Robust project management with the tilted beta distribution

market in the United States would cost them their own jobs, and fewer still would have

predicted that it would have global repercussions. Hence there is a growing appreciation

of the need for more robust models that assign greater probability to more extreme

events.

Recent research by authors such as Hahn (2008) and Lopez Martın et al. (2012)

has described some ways for increasing the amount of distributional uncertainty in the

context of project management tools such as PERT. The goal of this research stream

has been to extend the PERT framework to accommodate greater likelihood to extreme

tail-area outcomes. This has led to the ability to provide wider confidence intervals

for activity and project duration times and hence more conservative results, while still

retaining the classic PERT results as an important special case. The ability to increase

distributional uncertainty is an important first step towards robust project management

estimation; however, one consideration that has been underexplored is that one extreme

may be more likely or more important than another. For example, as documented below,

project managers tend to provide positively biased time estimates. Accordingly, project

management tools which depend on these biased estimates are likely to underpredict the

overall project time. In the current paper, we describe a new distribution that can be used

by an independent agent (such as a risk manager) to differentially weight high versus

low extremes. This can be used to help counteract some biased estimates.

This paper is structured as follows. Firstly, we review the literature about alternative

distributions used in the area of PERT methodology. In Section 3 we present the tilting

distribution, as a particular case of generalized Topp and Leone distribution; the tilted

beta distribution, as a mixture between the tilting and beta distributions, and some

stochastic characteristics. The elicitation for the distribution is presented in Section 4.

The results are illustrated with an example in Section 5. Finally, Section 6 summarizes

the main conclusions.

2. Literature

Projects often fail to meet various financial and scheduling targets despite management’s

best efforts to ensure success. For example, Bevilacqua et al. (2009) report on bud-

get overruns and non-completion of tasks in projects undertaken with the use of PERT

methodologies in the energy sector. Hence, there have been numerous studies which

have tried to understand the sources of the persistent problem of project management

overestimates or underestimates. Boulding et al. (1997) find that senior level executive

subjects tended to ignore negative information or distort the information to fit precon-

ceived notions and decisions. Hill et al. (2000) find that expert project managers some-

times overestimate and sometimes underestimate project durations, but that the under-

estimates were greater in magnitude leading on average to underestimation. Keil et al.

(2007) conducted a laboratory experiment which revealed that failure to recognize prob-

lems early also leads to over-optimistic assessments regarding information technology

Eugene D. Hahn and Marıa del Mar Lopez Martın 255

projects. Snow et al. (2007) describe an in-depth research program on the assessment

of biases among software development project managers. The most commonly reported

reason for giving optimistic judgments was to avoid being the bearer of bad news. Of the

56 surveyed project managers, 22% mentioned providing optimistic judgments because

of a belief that senior management “shoots the messenger” while another 22% indicated

that optimistic judgments were provided so as to make the project manager look good.

Project managers were also two times more likely to be optimistically biased than they

were to be pessimistically biased. Snow et al. (2007) conclude “optimistic bias leads

to status reports that are very different from reality, while pessimistically biased status

reports tend to be accurate because bias offsets error”. Iacovou et al. (2009) also found

that optimistically-biased reports were more prevalent than pessimistically biased ones

in a sample of 390 information systems project managers, a finding consistent with work

by Smith et al. (2001) and Gillard (2005). In a related vein, project managers who are

able to accurately assess the risks of a troubled project are more likely to discontinue the

project (Keil et al., 2000). Sengupta et al. (2008) conducted research on several hundred

project managers and found that managers seem to strongly anchor on the initial risk

assessment, and find it difficult to update their opinion with new information that should

have prompted a re-assessment. One of the mitigation strategies identified by Sengupta

et al. (2008) was better calibration of forecasting tools to project particulars.

Given the large volume of research which indicates project managers may tend

toward having an optimistic bias, one possible solution is to provide a system whereby

a third party (such as a risk manager) can provide an outside independent review to

help remove bias in estimates. Oztas and Okmen (2005) describe a project management

methodology called the judgmental risk analysis process that is explicitly pessimistic

in nature. This is implemented by assessing a separate risk factor for each activity

and assigning a probability distribution to the risk factor. The minimum and maximum

activity times are then modified by including additive and subtractive offsets based on

the activity risk factor to these activity times. Here we observe that if managers or other

experts tend to overemphasize optimistic information, then counteracting this is a matter

of de-emphasizing or downweighting optimistic information. In the current paper, we

provide a probabilistic approach that permits a risk manager to introduce a negative (or

positive) weighting across the range that nonetheless retains the usual PERT framework

as a special case. This is accomplished by the introduction of a new distribution called

the tilted beta distribution. In the following section, we present this distribution and

explore some of its main properties which are relevant for project management. We then

present an application of this distribution and conclude with discussion.

3. Distributions for project activity times

Malcolm et al. (1959) were the first to use the beta distribution to describe project man-

agement activity times. The beta distribution is the most prevalent distribution used in


stochastic project management due to its useful properties and appearance in the seminal

work of Malcolm et al. (1959). Other widely used continuous probability models within

the PERT methodology are the triangular distribution (Clark, 1962; MacCrimmon and

Ryavec, 1964; Moder and Rodgeres, 1968; Vaduva, 1971; Megill, 1984; Williams,

1992; Keefer and Verdini, 1993; Johnson, 1997), the trapezoidal distribution (Pouliquen,

1970; Herrerıas and Calvete, 1987; Herrerıas, 1989; Powell and Wilson, 1997; Garvey,

2000), the doubly truncated normal distribution (Kotiah and Wallace, 1973), the uniform

distribution (Suarez, 1980; Romero, 1991), the generalized biparabolic distribution

(Garcıa et al., 2010) and the Parkinson distribution (Trietsch et al., 2012).

More recently, the literature on distributions for project management activity times

has emphasized the importance of accounting for heavy tails and assigning more proba-

bility density to extreme values (Mohan et al., 2007). In addition, the research emphasis

has moved away from using ‘off the shelf’ statistical distributions and instead has sought

to engineer new distributions that are tailored to satisfy PERT desiderata. For example

Hahn (2008) proposed the beta rectangular distribution which is a bounded distribution

like the beta but assigns greater density to extreme values and can accommodate very

heavy tails. Similarly, Garcıa et al. (2010) presented the generalized biparabolic distri-

bution and demonstrated its capacity to have larger variances than the beta distribution.

Our motivation in writing this paper is to engineer a distribution to address the docu-

mented optimistic bias discussed above while also addressing the need for heavy tails

and large variances which has been identified previously.

3.1. Beta distribution

We begin with a brief review of the beta distribution given its importance in both the

current work and in the project management literature. The standard beta distribution

defined on [0,1] has the following probability density function (pdf)

p(x|α,β) =

Γ(α+β)

Γ(α)Γ(β)xα−1(1− x)β−1 if 0 ≤ x ≤ 1,

0 otherwise.

(1)

It is necessary that both α > 0 and β > 0 for (1) to be a valid pdf. The mean of (1) is

α/(α+β) while the variance is (αβ)/(

(α+β)2(α+β+1))

.

The beta distribution is capable of a variety of shapes (see distributions having dotted

lines in Figure 3). Unfortunately the beta distribution does not provide a great deal of

flexibility when it is of interest to preserve the typically-preferred unimodal shapes but

assign higher probability to extremal (or ‘tail area’) events. This observation led Hahn

(2008) to propose the beta rectangular distribution, which is a mixture of the beta and

rectangular distributions, for project management activity times and it is defined by


p(x|α,β ,θ) =

θΓ(α+β)

Γ(α)Γ(β)xα−1(1− x)β−1+(1−θ) if 0 ≤ x ≤ 1,

0 otherwise.

(2)

where θ is a mixing parameter such that 0 ≤ θ ≤ 1.

Under the PERT conditions (to be discussed in Section 4), the beta rectangular

distribution permits larger variances than the beta distribution and allows for elevated

tail-area density (see Figure 1. The beta rectangular also has the beta distribution as a

special case; hence, the classic PERT activity time parameters can be easily obtained as

a particular case.

0.2 0.4 0.6 0.8 1.0x

0.5

1.0

1.5

2.0

Density

0.2 0.4 0.6 0.8 1.0x

0.5

1.0

1.5

Density

α= 2 and β = 4 α= 3 and β = 3

Figure 1: Examples of beta rectangular distribution for θ = 1 (solid), θ = 0.8 (dotted), θ = 0.6 (dashed)

and θ = 0.4 (dash-dotted).

However, the previous discussion of Section 2 indicates that project managers may

have an optimistic bias and the beta rectangular does not provide a way for addressing

this issue. The remainder of this section is dedicated to formulating a distribution that

addresses this issue. Accordingly next we describe the tilting distribution which allows

for a straightforward way of expressing an optimistic (or pessimistic) bias. This will in

turn allow us to construct the tilted beta distribution whereupon we will study in depth

the characteristics of this distribution.

3.2. Tilting distribution

Topp and Leone (1955) present a distribution with probability density function (pdf)

defined by f (x,β) = β(2−2x)(2x− x2)β−1, where x ∈ [0,1] and β > 0. Depending on

the values of β the distribution either has a J-shaped form (0 < β < 1); is unimodal

(β > 1); or is left-triangular (β = 1). Kotz and van Dorp (2004) present a generalization

of the Topp and Leone distribution by considering a slightly more general generating

pdf, whose expression is


g(x|α) = α−2(α−1)x, (3)

where α defined on the interval [0,2]. The authors define the slope distributions as the

distributions with pdf of the form (3).

Taking the reparametrization α = 2v, we introduce a distribution, called the tilting

distribution, which has the following density function:

p(x|v) ={

2v−2(2v−1)x if 0 ≤ x ≤ 1,

0 otherwise.(4)

As 0 ≤ α ≤ 2, the parameter v is defined on the interval [0,1]. The reparameterization

considered here leads to a parameter range consistency for the tilted beta, as shown later.

The cumulative density function (CDF) of (4) is

F(x|v) =

0 if x < 0,

2vx− (2v−1)x2 if 0 ≤ x ≤ 1,

1 if x > 1.

(5)

Graphical examples of density and cumulative density function of the tilting distribution

are shown in Figure 2. Figure 2 reveals that when v = 1/2 the uniform distribution is

obtained and when v = 0 or v = 1, a triangular distribution with mode v is obtained.

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

CD

F

0.2 0.4 0.6 0.8 1.0

0.5

1.0

1.5

2.0

PD

F

x x

v = 0 v = 1/4 v = 1/2 v = 3/4 v = 1

Figure 2: Examples of PDF and CDF of tilting distributions.

The mean, variance and coefficient of skewness of the tilting distribution are respec-

tively

E(X) =2− v

3, var(X) =

2v(1− v)+1

18, β1 =

2√

2(

1+3v−15v2+10v3)

5(1+2v−2v2)3/2. (6)


Taking the first derivative of var(X) with respect to v the variance of the distribution is

maximized for v = 1/2. When v = 0 or v= 1 the variance of the distribution is minimum

whilst the coefficient of skewness is maximum.

In some contexts it can be necessary to work with a variable defined over the more

general support [a,b] instead of [0,1]. For cases where the variable may take on an

arbitrary location and scale, we describe the variable Y = a+(b−a)X with b > a. The

inverse function is X = y−a

b−awith Jacobian ∂X/∂y = 1

b−a. Then, the density function of

(4) with to support [a,b] is

p(y|w,a,b) = 2

b−a

w−a

b−a−(

2w−a

b−a−1

)(

y−a

b−a

)

if a ≤ y ≤ b,

0 otherwise,

(7)

where w = a+(b−a)v. The quantile function of Y is

P−1(q|w) =

a(2b−w)−bw+(b−a)√

a2(1−q)+b2q−2aw(1−q)−2bwq+w2

a−2w+bif w 6= a+b

2,

a+(b−a)q if w = a+b2,

(8)

with 0 < q < 1.

Although the introduction of additional parameters is associated with an increased

complexity for the distributional expressions, the increase in flexibility makes it worth-

while to briefly summarize a few key expressions. In this case, the mean and variance

of the tilting distribution are

E[Y ] =2a−w+2b

3, var[Y ] =

a2 +2(a+b)w+b2

6− (a+w+b)2

9. (9)

3.3. Tilted beta distribution

Having presented a few key properties of (4), we now introduce the tilted beta distribu-

tion. The density function of a random variable X having the tilted beta distribution with

α> 0, β > 0, v ∈ [0,1], and θ ∈ [0,1] is

p(x|v,α,β ,θ)=

(1−θ)[

2v−2(2v−1)x]

+θ[

Γ(α+β)Γ(α)Γ(β)x

α−1(1− x)β−1]

if 0 ≤ x ≤ 1,

0 otherwise.

(10)

This distribution can be seen as a mixture of the tilting and beta distribution. The

parameter θ indicates the relative proportionality of the tilting distribution to the beta

distribution and v can be interpreted as the relative tilt proportionality. When θ = 1


the beta distribution is obtained, for θ = 0 we obtain the tilting distribution of (4) and

θ = 1/2 indicates a balance between the two distributions. With respect to the parameter

v, v = 0 indicates the maximum downward tilt, v = 1 indicates maximum upward tilt,

and v = 1/2 indicates a balance of upward and downward tilt. Figure 3 shows that the

beta distribution, the uniform distribution, and the beta rectangular (Hahn, 2008) are

all special cases of the distribution (10). Indeed, the density of the resulting tilted beta

keeps the property of smoothness possessed by the beta distribution. This property can

be contrasted with discontinuous or ‘sharp’ distributions that have been proposed for

PERT such as the triangular (Johnson, 1997) and its extensions in the two-sided power

distribution family (Garcıa Perez et al., 2005; Herrerıas et al., 2009).

0.0 0.2 0.4 0.6 0.8 1.0x

0.5

1.0

1.5

2.0PDF

0.0 0.2 0.4 0.6 0.8 1.0x

0.5

1.0

1.5

2.0PDF

0.0 0.2 0.4 0.6 0.8 1.0x

0.5

1.0

1.5

2.0PDF

0.0 0.2 0.4 0.6 0.8 1.0x

0.5

1.0

1.5

2.0PDF

0.0 0.2 0.4 0.6 0.8 1.0x

0.5

1.0

1.5

2.0PDF

0.0 0.2 0.4 0.6 0.8 1.0x

0.5

1.0

1.5

2.0PDF

A B C

D E F

θ = 0 θ = 1/4 θ = 1/2 θ = 3/4 θ = 1

Figure 3: Examples of tilted beta distributions for: α= 2,β = 3,v = 0 (A); α= 2,β = 3,v = 0.5 (B),

α= 2,β = 3,v = 1 (C), α= 3,β = 2,v = 0 (D); α= 3,β = 2,v = 0.5 (E), α= 3,β = 2,v = 1 (F).

The moment generating function of (10) is defined by

Mx(t) = 2et (t −1+(2− t)v)+1− (2+ t)v

t2(1−θ)+ 1F1[α,α+β , t]θ (11)

where 1F1 indicates the Kummer confluent hypergeometric function. From (11) one can

obtain the mean and the second moment of the tilted beta distribution

E(X) = (1−θ)2− v

3+θ

α

α+β, (12)

E(X2) = (1−θ)3−2v

6+θ

α(α+1)

(α+β)(α+β+1). (13)


We can consider the density of the pdf (10) at the endpoints making the usual

assumption that α and β are both greater than or equal to 1. When x = 0, it can be

shown that p(x|v,α,β ,θ) = (1−θ)2v. When x = 1, p(x|v,α,β ,θ) = (1−θ)(2− 2v).

Observe that the density at the endpoints will be different as long as v 6= 0.5 and θ 6= 1.

As will be mentioned later, very few distributions applied in the PERT methodology

have this property.

In the existing literature on this issue, few distributions exhibit a shape similar to

that of the tilted beta. One such distribution is the elevated two-sided power distribution

introducing by Garcıa et al. (2011). However, here we respectfully note that the elevated

two-sided power distribution requires an additional parameter and possesses more com-

plex expressions (cf. the simplicity of (13) above versus equation (23) of Garcıa et al.

(2011)). In contrast, the current distribution is a mixture of two tractable distributions

and hence it is straightforward to implement in any environment where the beta distri-

bution of PERT has been previously applied. The reflected generalized Topp and Leone

distribution (Van Dorp and Kotz, 2006) can achieve somewhat similar shapes but it does

not seem possible for this distribution to have appreciable density at both extremes si-

multaneously. Thus, this distribution is not well-suited for circumstances where tail-area

events have appreciable likelihood at both high and low extremes. Moreover, Van Dorp

and Kotz (2006) indicate that this distribution does not have closed-form moment ex-

pressions except for special cases (cf. with (13) above), again adding computational

cost for applications-oriented Monte Carlo simulation. Finally, Pham-Gia and Turkkan

(1993) presented an explicit expression for the distribution of the difference of two beta

distributions. This distribution can also take on many flexible shapes (Nadarajah and

Gupta, 2004, see pp. 71–84). However, it also has a complex specification and, for ex-

ample, its moments can only be analytically approximated.

Note that the procedure to raise any bounded continuous distribution by the tilting

distribution is equivalent to the procedure of raising the density of the distribution

linearly, and then re-normalizing.

4. Elicitation of the tilted beta distribution’s parameters

In many project management applications, it is necessary to consider parameter elic-

itation for distributions. Direct elicitation of the beta distribution’s α and β is always

an option (e.g., Chaloner and Duncan, 1983). Historically however project management

applications have used the classic PERT parameters: a (lower bound), m (most likely)

and b (upper bound). The classic PERT formulas are then

E(Y ) =a+4m+b

6, (14)

V(Y ) =(b−a)2

36. (15)


A wide literature has been dedicated to examining the necessary conditions linking

(14) and (15) to the parameters of the beta distribution (Malcolm et al., 1959; Clark,

1962; Grubbs, 1962; Sasieni, 1986; Gallagher, 1987; Littlefield and Randolph, 1987;

Kamburowski, 1997). To summarize, (14) holds exactly when k = α+β = 6 and α 6= β .

We may call this the Type I beta condition. Further, (14) and (15) simultaneously hold

when: α = β = 4; α = 3+√

2, β = 3−√

2; and α = 3−√

2, β = 3+√

2 (Grubbs,

1962). We may call this the Type II beta condition. Clearly the Type II condition is

more restrictive than the Type I condition. In this case, all that is required is to select

whether a symmetric, positively skewed, or negatively skewed distribution is required.

Then the values of α and β are given as above. For the Type I condition, note that

the mean and mode of the beta distribution are α/k and (α− 1)/(k− 2), respectively.

Hence, solving simultaneous equations for the mean and the mode we find in the case

of a standardized beta (a = 0 and b = 1) that the values of α and β under the Type I

condition are α= 4m+1 and β = 5−4m.

Having addressed the elicitation of α and β , we turn to the elicitation of the

remaining parameters of (10). The elicitation of the mixture parameter θ has been

considered by Hahn (2008) and Lopez Martın et al. (2012) using the parameter λ. Hence

it remains to discuss v. Eliciting v can be accomplished by the following procedure. We

assume the expert believes there is a linear increase or decrease in the probability density

across time in accordance with the shape of the tilting distribution. Let A j represent the

event that a particular activity is completed on day j. Then we ask the expert to provide

the probability of the event of activity completion in day j. This is denoted by p(A j).

Next we ask her to give the probability of the event of activity completion in day j+1,

which is denoted by p(A j+1). Suppose a discrete approximation to the tilting distribution

is used. The slope is (see Figure 4)

p(A j+1)− p(A j)

( j+1)− j(16)

j0 1j+1

2v−2(2v−1)x

p(A j)

p(A j+1)

Figure 4: Cumulative distribution function of a discrete variable with support [0,1].


by the definition of the slope asy2−y1x2−x1

. Since there are b − a conceivable activity

completion days, we may normalize the cumulative activity time until A j+1 as x2 =j+1

b−a. Similarly we may normalize the cumulative activity time until A j as x1 =

j

b−a.

Since A j+1 and A j differ by one day out of the b− a total activity days, we substitute

into the slope formula to define the rate of change as

r =p(A j+1)− p(A j)

1/(b−a). (17)

Note that other time units besides days may be alternatively used. Once we have

obtained r, we can solve for the value of v by making r equal to the slope of the density

function −2(2v−1) and solving in terms of v, yielding

v =2− r

4. (18)

If v 6∈ [0,1], then a re-examination will be required. Discussion with the expert can

be undertaken to reveal whether, for example, the judgment task can be made easier

by considering months instead of days. Alternatively, it may be that a linearly-sloped

distribution does not correspond to the expert beliefs and if so the process would need to

be terminated. Assuming a valid value of v, conversion to w is given by w= a+(b−a)v.

Alternatively, we ask the expert the probability of the event of activity completion

in day j+1, j+2, . . . , j+ k, where the period j+1 is the following day of the first day

after the start of the project and j+ k is the day before the end of the project. For each

probability, and using the expression (18), we elicit the parameter v for each different

day. For example we obtain v1,v2, . . . ,vk. We can then find v as the arithmetic mean

v = 1k ∑

ki=1 vi.

Another more informal approach to elicitation of v can be contemplated by analogy

with the elicitation of θ . Note in the beta rectangular distribution that θ = 0 corresponds

to the case of no (or 0%) additional uncertainty above and beyond that of the beta.

Further, θ = 1 corresponds to the case of complete (or 100%) uncertainty. So v = 1

would correspond to a 100% linear pessimistic belief or worst-case linear belief about

the project activity completion time. In contrast, v = 0 would correspond to a 0% linear

pessimistic belief (100% linear optimistic belief) or a best-case linear belief about the

project activity completion time. More moderate values of v would represent various

compromises between these extremes, with v = 1/2 representing neither pessimism nor

optimism. In the event that we would want to counteract an elicited value v, we simply

invert the slope by using (1− v) in the place of v.


5. Application

Figure 5 shows 29 activities in a real-world electronic module development project from

Moder et al. (1983). The critical path is marked by a heavy line.

Receive and

test preamp

4719

Let contract

for preamp

4718

6-8-16

14-17-32

2-4-8

14-17-32

2-3-5

0

0

0

0

1-2-3

0.4-0.6-1 2-3-5 3-4-5 8-10-12

2-3-5 3-4-5 4-6-8

6-8-16

6-8-12

6-8-123-4-5

2-4-5

1-2-3

1-2-3

0.1-0.5-1

1-2-4

1-2-3

1-2-3

3-4-

5

3-4-5

2-3-

5

1-2-5

1-2-30.4-0.6-1

Let contract

for Smith

trombone

4715

Dummy

start event

0001

Let contract

for preamp

prototype

4725

Receive Smith

trombone

and begin

test 4716

Receive Jones

trombone

and begin

test 4713

BBC

linkage

4431

BBE

structure

0206

Start

microwave

structure

0206

Paramp

prototype

test

4726

Begin choice

of trombone

4714

Requests for

!ight hdwre.

proposals

4731

Start "nal

mech.

design

4727

Trombone

for systems

test

4322

Request

!ight

hardware

paramp

proposals

4728

Let

preamp

contract

4732

Let

trombone

contract

4735

Let

mixer

contract

4737

Receive

preamp

and begin

tests 4733

Receive

trombone

and begin

tests 4736

Receive

mixer

and begin

tests 4738

Receive

hardware

and begin

tests 4741

Let L.O. and

power-split

contract

4740

Let L.O. and

Ref. power-

split contract

4739

Let

paramp

contract

4729

Receive

paramp

and begin

test 4730

Start choice

of preamp

vs paramp

4734

Begin

microwave

assembly

4742

BBE

receive

microwave

support 0621

Start "nal

assembly

test

4743

Complete

test and

deliver to

system

41990.4-0.6-1

Project activities and paths

Figure 5: Reproduction of PERT network of an electronic module development project (Moder et al.,

1983).

The distribution for the total project time can be found using Monte Carlo simulation

and accounting for the diagram’s precedence relationships. To obtain our results, we

simulated from the activity times using the beta distribution information from Figure 5

and the listed values of θ and v appearing in our results. The results are based on 10000

Monte Carlo simulations from the distributions of interest. Please observe that we report

results arising from use of the beta rectangular distribution and, for completeness, use

of the tilted beta distribution with v = 1/2 which is equivalent to the beta rectangular.

Results for the two equivalent distributions are equivalent up to Monte Carlo error at

the third significant digit with a few exceptions that are slightly larger such as the upper

95% confidence interval for θ = 1/4 in Table 1 (62.34 versus 62.13).

With increasing θ , the distributions approach the beta distribution and they equal the

beta when θ = 1. Therefore we see in Table 1 that the standard deviation declines with


Table 1: Stochastic characteristics of the total project time variable obtained by Monte Carlo simulations

where Beta-R is the beta rectangular distribution and T-Beta is the tilted beta distribution.

θ Distribution Mean Stand. Dev. Skewness Kurtosis 95% C.I.

θ = 1/4

Beta-R 51.50 5.55 0.23 2.26 (42.21, 62.34)

T-Beta (v = 0) 56.56 5.42 -0.17 2.15 (46.41, 65.49)

T-Beta (v = 1/4) 54.08 5.65 0.02 2.11 (44.13, 64.10)

T-Beta (v = 1/2) 51.51 5.56 0.20 2.22 (42.12, 62.13)

T-Beta (v = 3/4) 48.85 5.14 0.46 2.58 (40.64, 59.80)

θ = 1/2

Beta-R 50.15 5.09 0.47 2.54 (42.05, 60.96)

T-Beta (v = 0) 53.54 5.49 0.21 2.14 (44.38, 63.92)

T-Beta (v = 1/4) 51.86 5.38 0.36 2.30 (43.17, 62.59)

T-Beta (v = 1/2) 50.11 5.10 0.48 2.54 (42.18, 60.88)

T-Beta (v = 3/4) 48.54 4.78 0.54 2.77 (41.03, 59.12)

θ = 3/4

Beta-R 48.88 4.48 0.67 3.10 (41.96, 59.33)

T-Beta (v = 0) 50.56 4.90 0.60 2.74 (43.09, 61.51)

T-Beta (v = 1/4) 49.82 4.70 0.65 2.93 (42.63, 60.57)

T-Beta (v = 1/2) 48.87 4.44 0.69 3.17 (41.95, 59.23

T-Beta (v = 3/4) 48.09 4.21 0.60 3.07 (41.44, 57.72)

increasing θ . This is because when the tilting distribution predominates, the dispersion

is increased which makes estimates wider and more conservative. For v, higher values

correspond to assigning more weight to shorter, more optimistic outcomes. Accordingly

the means in Table 1 are monotonically decreasing in v. It is also somewhat surprising

to note that the standard deviations also are decreasing in v (except for the case when

v = 0 and θ = 1/4). Inspection of Figure 5 indicates that the judgments rendered tend

to be optimistic or neutral at worst (coincidentally, this is consistent with our review in

Section 2). Hence a value of v = 1/4 further concentrates the optimistic nature of the

judgments into the shorter times, reducing the standard deviation. Larger values of v

counteract this, especially when θ is low.

Graphical displays of the distributions of total project times appear in Figure 6. The

most conservative results for the distribution of total project time can be seen when

v = 1/4 and θ = 1/4. This distribution is the least skewed of all those displayed, and

appears approximately uniform across the middle third of its range. We observe that

the distribution in the centre of the top row for the beta rectangular when θ = 1/2 is

equivalent to the distribution for the tilted beta with θ = 1/2,v = 1/2 in the centre of

the third row, and these coincide as we would expect.

Figure 7 provides another way of viewing the changes in project times as a function

of distributional parameters. It plots the CDF for the simulated project times under

selected values of θ and v. On the left side where θ = 1/4, the tilting portion of the

mixture is predominant. We see there that the optimistic assessment of v = 3/4 leads to


Beta rectangular θ = 1/4. Beta rectangular θ = 1/2. Beta rectangular θ = 3/4.

Tilted beta θ = 1/4,v = 1/4. Tilted beta θ = 1/2,v = 1/4. Tilted beta θ = 3/4,v = 1/4.



Figure 6: Distributions of the total project time (Electronic Module Development Project).

a relatively high cumulative probability of completion by 55 days. For less optimistic

values of v, the cumulative probability of completion by 55 days (or other values we

might select) falls off considerably. The right side of Figure 7 shows the CDF when the

beta portion of the mixture predominates. The CDFs have some variability due to v but

in general the CDFs are closer together and rise more steeply since they preserve more

of the classic PERT beta influence. For completeness, we also observe that the CDF for

the beta rectangular and the tilted beta with v = 1/2 again gives essentially the same

result, as we would expect, since the solid beta rectangular line and the dashed tilted

beta v = 1/2 line are essentially superimposed.


1.0

0.8

0.6

0.4

0.2

0.035 40 45 50 55 60 65 70

days

1.0

0.8

0.6

0.4

0.2

0.035 40 45 50 55 60 65 70

days

T-Beta (v = 0)

T-Beta (v = 0)

T-Beta (v = 1/4)

T-Beta (v = 1/4)

Beta-R (solid line)

Beta-R (solid line)

T-Beta (v = 1/2)

T-Beta (v = 1/2)

T-Beta (v = 3/4)

T-Beta (v = 3/4)

A B

Figure 7: Cumulative distribution function of beta rectangular distribution and tilted beta distribution for

θ = 1/4 (A) and θ = 3/4 (B)

Finally, Table 2 shows an example of what would constitute a rare or tail-area event

under the different distributions. Here the 95% value of the CDF is obtained for v and

θ taking on the values 0, 1/4, 1/2 and 3/4. With the introduction of the parameter v, the

CDF of the tilted beta distribution may provide project time-to-completion estimates that

can be either higher or lower than the beta rectangular distribution. The most striking

comparison involves in the case when θ = 1 which is the classic PERT beta case. We

observe the classic PERT result would say there is a 95% chance of the project being

completed by approximately 53.9 days, excluding some Monte Carlo error visible for

the four values of v in the plot. Compare this results with the case of θ = 3/4 where

a small amount of extra-beta variability has been mixed in but the beta distribution

still predominates at θ = 3/4. Here even under the most optimistic case (v = 3/4) the

95% completion time has increased by 2 days to about 55.9 days. Hence, this example

‘worst-case scenario’ is two days worse than that given by the classic PERT beta. With

less optimistic values of v, the time increases further. For example, with v = 3/4 and

θ = 1/2 we approach 57.7, i.e. approximately four days more than PERT.

Table 2: The maximum time needed to complete the 95% the project for beta rectangular (Beta-R) and

tilted beta (T-Beta) with v = 0, v = 1/4, v = 1/2 and v = 3/4.

θ Beta-R T-Beta (v = 0) T-Beta (v = 1/4) T-Beta (v = 1/2) T-Beta (v = 3/4)

0.25 60.98 64.68 63.06 60.95 58.39

0.50 59.58 62.80 61.45 59.60 57.69

0.75 57.75 60.03 59.22 57.68 55.98

1.00 53.90 53.92 53.69 53.78 53.83

In most cases, when the parameter v is higher than 1/2, the tilted beta distribution

will be by construction more optimistic than the beta rectangular distribution and as a


consequence its estimates will indicate that a lower time will be needed to complete the

project. Conversely, when v is lower than 1/2, the tilted beta distribution provides more

conservative estimations.

6. Conclusion

The introduction of different activity distributions has played an important part in the

PERT methodology. However, this issue divides the researchers of this field. Some

authors argue against the introduction of new probability distributions into PERT (see

Clark, 1962; Hajdu, 2013; Hajdu and Bokor, 2014). Conversely as shown in Section 3

other authors have applied new distributions. Regarding the current paper, this debate

has parallels in statistical practice. Some authors use robust statistical methods to handle

outliers while other authors adopt less formal techniques or may even naively do nothing

at all.

This paper introduced the tilted beta distribution and shown how it can be used in

project management. Since the classic PERT results can be reproduced, it is simple to

adopt in any environment where PERT is utilized. We can easily explain to executives

and decision makers that incorporating additional uncertainty can help us to arrive

at new insights. Elicitation of parameters is straightforward or one could perform

sensitivity analysis using several parameter values as we have done here. The tilted beta

has a number of attractive computational properties such as being easy to simulate from

and having closed-form moment expressions. In summary, the tilted beta distribution

provides project managers with a flexible and easy to work with distribution that allows

for the extensive representation of optimistic or pessimistic beliefs regarding activity

times.

Past work has pointed out the need to describe a more flexible distribution which al-

lows for varying amounts of dispersion and greater likelihoods of more extreme tail-area

events (Hahn, 2008). The construction of beta rectangular distribution is characterized

by greater flexibility in the variance. However, this distribution assigns the same prob-

ability density at both the high and low extreme values. With the introduction of tilted

beta distribution, we have expanded the set of continuous type distributions defined on

a bounded domain, with the advantage of accommodating different relative likelihoods

of high versus low extreme tail-area events and, as opposed to other distributions ap-

plied in this methodology, the tilted beta has an expression of the expected value where

the extreme values have different weight. As a consequence, this distribution will be

more relevant for modeling a broader range of heavy tailed phenomena. In a closely re-

lated work, the elevated two-sided power distribution (Garcıa et al., 2011) also permits

different relative likelihoods of high versus low extreme tail-area events; however, as

described above, the current distribution is simpler to use in practice. Furthermore, note

that the procedure to induce tilting can be applied to any bounded continuous distribu-

tion.


We have compared the results of the tilted beta distribution with the results of beta

rectangular distribution for different values of the parameter θ and v. The results of

the application show that this probabilistic model permits a risk manager to incorporate

more optimistic and pessimistic scenarios than the beta rectangular distribution due to

the flexibility of the tilted beta. Our literature review suggests that experts may tend to

be too optimistic, and the beta distribution gives little weight to outliers in the standard

PERT Type I and Type II beta conditions. The current methodology redresses these

issues.

The distributions presented in this paper will be closer to the uniform since they

give more weight to the tails. We think this gives some evidence that the distribution

chosen is relevant by considering a recent paper by Hajdu and Bokor (2014). For larger

projects, 10% can be the difference between a project being on time or late and the

authors show that the uniform distribution is similar to PERT-beta + 10%. Furthermore,

for small projects the authors state that it may not matter much. However, the incentive

to use PERT with smaller projects is probably smaller.

There are at least two areas in which this research can be extended: first, the use of

heavy-tailed distributions in the context of different activity calendars (Hajdu, 2013);

second, to find more applications of these distributions fitting extreme tail-area events

which are present in a great variety of fields such us finance, groundwater hydrology

and atmospheric science among others.

Acknowledgements

We would like to thank referees for their detailed, encouraging and constructive reviews

of our paper which clearly contributed to improving both the structure and the content

of this manuscript.

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Robust project management with the tilted beta distribution